I agree with @maxbiostat that the model as you describe it is a bit weird so maybe there is a better solution if you describe the problem more fully. However, provided you are sure this is a model you want to implement, it doesn’t appear there is some nice analytical solution to marginalize N out completely, so some sort of partial-sum as you’ve proposed might be the best one can do.
Unfortunately, the partial sum will likely be costly unless \lambda (and hence the sensible range for N) is small. The Poisson-binomial is itself a distribution that is quite costly (at least quadratic in N) to compute (see Poisson-binomial distribution - any existing Stan implementation? - #7 by Bob_Carpenter if you haven’t already), so the model might be very slow…
It should however be possible to use something like simulation-based calibration (e.g. via the SBC package I am helping to develop) against your full simulator to understand to what extent does the partial sum approximation skew your results away from the exact solution.
Best of luck with your model!